William Krasker is back at it now that the NFL season offers strategic decisions to analyze. His first installment analyzes Denver coach Mike Shanahan's decision to run another play at the Jacksonville 24 yard line, rather than kicking a field goal, when trailing by 2 points with 37 seconds left. Denver fumbled, effectively ending the game. Of course, since the call blew up on him, Shanahan has faced the predictable media criticism.
Note that when faced with strategic decisions, the best choice is generally the one that maximizes the probability of winning the game. With that in mind, here is Krasker's analysis:
From data on all FG attempts during the 2003 regular season and playoffs, which we described in a previous article, it appears that NFL place kickers make about 76% of their field goals from 41 yards. Moreover, the success probability rises about 1% per yard as the distance decreases. Therefore, to a first approximation, a play that (expectationally) gains 3 yards increases the probability of making the FG from 0.76 to 0.79. So, if we let pf denote the probability of losing a fumble, we find that Denver's probability of taking the lead is (1 − pf )0.79 if they run another play, versus 0.76 if they kick immediately. It's hard to see how the probability of losing a fumble can be more than about 0.015. (As a point of reference, in the last four seasons Tiki Barber lost 17 fumbles in 961 carries.) So Denver's probability of taking the lead is about 0.778 if they run another play. Moreover, since Denver's probability of keeping the lead is greater if they use extra time and force Jacksonville to use a timeout, we conclude that Mike Shanahan made the right decision.
It's a close call, but that's what coaches and managers are paid to do: make dozens of decisions in a game, each of which might involve just a few hundredths of a percent of incremental probability.
A coach that makes the right calls repeatedly - say 40 decisions per game with an average gain of .0025 in probability - has a significant cumulative impact on the outcome. By this standard, Mike Tice's tenure in Minnesota won't be very long. Check out the analysis - Tice threw away a big chunk of probability in Monday Night's game.